Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).

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1answer
80 views

Radius of Convergence and the Frobenius Method

Consider the equation $$4xy'' + 2y'+ y = 0$$ I know that $x=4$ is a regular singular point, and in the notation that my uni uses, we say that: $$(x-x_0)^2 y'' + (x-x_0)p(x)y' + q(x)y = 0$$ where ...
2
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1answer
23 views

Power series for inverse of truncated power series of $e$

Let $T_n(x)=\sum_{k=0}^{n}\frac{x^k}{k!}$. I'm looking at the function $$f(x)=\frac{1}{T_n(x)}$$ and I would like to find the power series of this particular function. I know I can use Maclaurin ...
1
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2answers
45 views

$\int_{-L}^{+L}h(z)\,dz = 2 \sum_{-\infty}^{+\infty}\frac {a_n}{n} \sin (nL)$

is it possible to find a formula for $a_n$ from $$\int_{-L}^{+L}h(z)\,dz = 2 \sum_{-\infty}^{+\infty}\frac {a_n}{n} \sin (nL)$$ For $n=0$ the series is $0$ Thanks
2
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1answer
37 views

Powers and Power Series'

When doing some problems I came across the function: $$f(x)=\frac{x}{1-2x}$$ I realised that the Maclaurin expansion of this function was: $$f(x)=x+2x^2+4x^3+16x^4...$$ Evaluate at $x=1$ to get ...
0
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4answers
39 views

Why does $\lim_{n \to \infty} \sum_{k=1}^n\frac{t^{k+1}}{(k+1)!}=e^t-t-1$?

Why does $\lim_{n \to \infty} \sum_{k=1}^n\frac{t^{k+1}}{(k+1)!}=e^t-t-1$? I know $\lim_{n \to \infty} \sum_{k=0}^n\frac{t^k}{k!}=e^t$, but my sum starts at $k=1$ and also has ...
4
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0answers
79 views

Calculate $1^1 + 2^2 + 3^3 + … + n^n$

Is there a formula to calculate $1^1 + 2^2 + 3^3 + ... + n^n$ I searched but didn't find a formula for increasing powers
0
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1answer
41 views

Using the Maclaurin series to approximate $f(0.1)$ for $f(x)=(3+e^{2x})^{0.5}$

I was tasked to use the Maclaurin series to calculate $f(0.1)$ of $f(x)=(3+e^{2x})^{0.5}$. I got the Maclaurin expansion of $p_2(x) = \sqrt{3} + 4x +5x^2$ into which I plugged $0.1$ to yield ...
1
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3answers
92 views

Series expansion of $\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$?

How would I find the series expansion $\displaystyle\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$ so that it will turn into an infinite power series again??
5
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4answers
287 views

Find the fraction where the decimal expansion is infinite?

Find the fraction with integers for the numerator and denominator, where the decimal expansion is $0.11235.....$ The numerator and denominator must be less than $100$. Find the fraction. I ...
0
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0answers
38 views

Truncation of partitions generating function question

$A (x)$ is the generating function for partitions. $B(x)=\sum_{n=0}^{\infty}b_nx^n $ $$b_n =\binom{\text{number of partitions of }n}{\text{into an even number of parts}}-\binom{\text{number of ...
4
votes
1answer
91 views

How to write $1-x-x^3+x^4+x^5+x^6-x^7 \cdots$ as a power series representation

How can I write $1-x-x^3+x^4+x^5+x^6-x^7 ....$ as a power series representation (i.e., a neat fraction such as $\frac{1}{1-x}$. This stems from $\binom{\text{number of partitions of }n}{\text{into an ...
2
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3answers
66 views

Writing $1+3x^2+8x^4+21x^6+\cdots$ as a power series representation

How would I write the power series $$1+3x^2+8x^4+21x^6+\cdots$$ as a power series representation (something neat similar to $\frac{1}{1-x}$)? This reminds me of the power series ...
4
votes
1answer
89 views

Adding Two Power Series if their bounds are different

I have the following product $$\frac{1}{6}\sum_{n=0}^{\infty}\left(\frac{-x}{2}\right)^n\sum_{n=0}^{\infty}\frac{a_nx^{n+2}}{n!}$$ Where $a_n$ is an arbitrary coefficient. If I factor out $x^2$ ...
2
votes
1answer
105 views

summation of a finite sequence?

What is the summation of the finite sequence: $$\sum\limits_{i = 1}^n {\frac{1}{i}\left( {\begin{array}{*{20}{c}} {2i - 2}\\ {i - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {2n + 2 - 2i}\\ ...
6
votes
4answers
126 views

How can I write this power series as a power series representation?

How can I write this power series ($1+x+2x^2+2x^3+3x^4+3x^5+4x^6+4x^7+5x^8....$) as a power series representation (like $\dfrac{1}{1-x}$ or something neat like that)?
2
votes
1answer
43 views

Determining a radius convergence of a power series

Let $$ \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1} $$ Is there an immediate way to determine $R=1$?
3
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1answer
42 views

Radius of convergence, prove that $\sum\limits_{n=0}^{\infty} a_n z^n$ converges absolutely for every $z \in \mathbb{C}$ with $|z| < R$

A proof of this is given in my lecture notes as follows: We define $R$ to be $\sup \{|z| \in \mathbb{R} : \sum |c_k z^k|$ converges $\}$ when the supremum exists. Prove that $\sum |c_k ...
3
votes
2answers
217 views

How to calculate the series?

How can we calculate the series: $$ F(x)=\sum_{n=1}^{\infty}\frac{(-1)^n}{1-x^n} $$ I found that $$ ...
3
votes
1answer
50 views

Writing solution to an arbitrary ODE with arbitrary initial values as the sum of a power series?

How can we solve for $y$ with these arbitrary initial values and polynomials? How would we write the solution as a power series?
1
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1answer
50 views

Can this infinite summation be simplified?

I encountered the following infinite summation $$\sum_{k=0,k\neq m}^{\infty}\frac{x^k}{(k-m)k!},x>0,$$ can it be simplified? Thanks!
2
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1answer
36 views

Find the leftmost (most significant digits) of a large exponent calculation, say $99^{99}$

I want to find the initial 10 digits of an exponent calculation whose result is a very large number - Say, $99^{99} = 3.697296 \times 10^{197}$ I only need to know the digits $3697296$ Is there any ...
1
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0answers
27 views

derivative of t distribution cdf wrt degrees of freedom

Given the cdf of a t distribution as follows: $T_\nu(x)=\frac{1}{2} + x\Gamma(\frac{\nu+1}{2}) + \frac{_2F_1 ...
1
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2answers
67 views

Questionable Power Series for $1/x$ about $x=0$

WolframAlpha states that The power series for $1/x$ about $x=0$ is: $$1/x = \sum_{n=0}^{\infty} (-1)^n(x-1)^n$$ This is supposedly incorrect, isnt it? This is showing the power series about ...
3
votes
2answers
65 views

Prove two identities relating to series

Show that: $$ (1). \sum_{n=1}^{\infty}\ln\big(\cos \frac{x}{2^n}\big)=\ln \frac{\sin x}{x} $$ $$ (2). \sum_{n=1}^{\infty}\frac{1}{2^n}\tan \frac{x}{2^n}=\frac{1}{x}-\cot x $$ Thank you in advance. ...
0
votes
4answers
44 views

Infinite Sum of 1/Polynomial

I'm trying to solve this equation: $$\sum_{k = 0}^{\infty}\dfrac{1}{(k+1)(k+3)}$$ Original image at http://i.imgur.com/wXZFxn0.png I attempted to find the sums of $\sum_0^∞\frac{1}{k+1}$ and ...
0
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1answer
28 views

Power Series Solutions And Minimum Radius of convergence [on hold]

Help with power series and minimum radius of convergence. Does the equation $$ (x^2 + 25)y'' + xy' + x^3y = 0 $$ have a power series solution $y = \sum_{n=0}^\infty c_n x^n$? If yes, ...
0
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0answers
48 views

Fourier series for logarithm of sine.

I looked up here: Fourier series of Log sine and Log cos I have modified the question: How can I derive the coefficient $a_n, b_n$ for $\log(\sin(x))$ in the fourier series representation? Also, I ...
0
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4answers
53 views

Power series for $f(x) = \frac{4}{x+2}$

Find the power series $f(x) = 4/(x+2)$ We know the geometric series: $$\sum_{n=1}^{\infty} x^{n-1} = \frac{1}{1-x}$$ $(x+2) = 1 - (-x - 1)$ So: $$\sum_{n=1}^{\infty} (-1)^{n-1}\cdot(x + 1)^{n-1} ...
0
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0answers
42 views

Remainder of $\ln x$ converges to $0$

I'm learning about power series and struggling to prove If $f(x)=\ln x$ prove that $R_n(f,c)(x)$ converges to $0$ where $c=1$. By some calculating I know that ...
0
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1answer
32 views

How to compute this series

I have to compute this series : $$\sum_{k=0}^\infty (k+1)x^{2k}$$ First, I have $$|x|<1$$ but then I don't know how to begin ...
0
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1answer
13 views

Get the closed form of Taylor series with Maple

Is it possible to get the closed form of Taylor series with Maple? The series command can give any given number of terms, but the question is about the closed form ...
2
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2answers
54 views

how to find convergence and divergence of the series [closed]

consider the following two series of complex numbers $$s_1=\sum_1^\infty\frac{i^{n}(2-\sin n)}{2^n.n}$$ $$s_2=\sum_1^\infty\frac{i^n(2-\sin n)}{2^n.n^2}$$ then a)both the series converge ...
3
votes
2answers
38 views

How to solve linear, second order ODE with Frobenius method with a difficult recurrence relation?

The ODE in question is: $$4xy''+2y'+y=0$$ Shifting the power series of each term so that they are all raised to the power $(n+r)$ will yield this recurrence relation: $$a_{n+1}={a_n\over ...
2
votes
0answers
43 views

Second order differential equation, power series method

Solve the differential equation $$(x+2)y''-xy'+(1-x^2)y=0 ; \quad X_0=1$$ using the power series method about the point $x_0=1$. I get to this step after deriving the derivatives of the ...
2
votes
2answers
381 views

Radius of convergence of product

Let $\sum_{i=0}^\infty a_nz^n$ and $\sum_{i=0}^\infty b_nz^n$ be power series, and define the product $\sum_{i=0}^\infty c_nz^n$ by $c_n=a_0b_n+a_1b_{n-1}+\ldots+a_nb_0$. Find an example where the ...
0
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2answers
35 views

to find radius of convergence of power series.

I have a power series given as: $f(z) =1 + z+ \frac{z^2}{2^2} +\frac{z^3}{3!} + \frac{z^4}{2^4} \frac{z^2}{2^2}+ \frac{z^5}{5!}+ \ldots$ I have to find radius of convergence of above series. My ...
0
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1answer
27 views

If each $a_n >0$ and $\sum a_n$ diverges, prove that $\sum a_n x^n \to +\infty$ as $x\to1^-$.

Exericise 9.36: If each $a_n >0$ and $\sum a_n$ diverges, prove that $\sum a_n x^n \to +\infty$ as $x\to1^-$. (Assuming $\sum a_n x^n$ converges for $|x|<1$. Here's what I tried: Since ...
1
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1answer
36 views

Show that $f(z):=\sum a_n (z-z_0)^n$ is continuous whenever $z$ is in disk of convergence.

Consider a power series $\sum a_n(z-z_0)^n$, and assume it has radius of convergence $r$. Then we know that $\forall z\in(z_0 -r,z_0 +r)$, this power series converges absolutely by root test. Thus we ...
7
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1answer
83 views

How to derive this interesting identity for $\log(\sin(x))$ [duplicate]

I saw on SE that: $$\log(\sin x)=-\log(2)-\sum_{n=1}^{\infty}\frac{\cos(2nx)}{n} \phantom{a} (0<x<\pi)$$ This is an extremely useful identity, as it helps solve: $$\int_{0}^{\pi} ...
1
vote
1answer
14 views

Taylor expansion of a logaritmic function

A function is given as $ln (y) = ln(\alpha)-\frac{\lambda}{\gamma}ln(\delta L^{-\gamma}+(1-\delta)K^{-\gamma})$ I need to find the second order Taylor $ln(y)$ around $\gamma=0$. How can it be done ...
2
votes
1answer
39 views

using power expansion to find limit

I am preparing for my final exam, and stuck on this question. Using power series expansion, evaluate $$\lim_{x\to 0} \frac{x\cos(x) -\sin(x)}{x^2-x\ln(1+x)}$$ I have no idea how to proceed. ...
4
votes
1answer
63 views

Expressing “formally” $f(x)=\frac {1}{\sqrt {1-2x}}$ as a power series

I have to express $f(x)=\frac {1}{\sqrt {1-2x}}$ as a power series and give its interval of convergence. Knowing the binomial series is as follows this should be fairly easy: $$(1+x)^{\alpha}=\sum ...
0
votes
2answers
64 views

Closed form of a series

Is there exist a closed form for the series of the form $$ \sum_{k=0}^{[n/2]}(-a)^{k}\binom{n-k}{k} $$ where $0<a\leq1$. For example, we have $$ ...
4
votes
1answer
33 views

power series for $\int_0^x e^{-t^2}dt$

Use a known power series expansion to find the power series representation of the integral function $g(x) =\int_0^x e^{-t^2}dt$ centered at $a=0$ My approach Note that $g'(x) = e^{-x^2}$. ...
0
votes
1answer
52 views

Multiplicative Inverse of a Power Series

For a formal power series $$F(x) = \sum p_i x^i$$ a multiplicative inverse of $F$ exists iff $p_0 \neq 0$. The inverse $\sum q_i x^i$ satisfies the recursion $$q_0 =\frac{1}{p_0}\\ q_{n} = ...
0
votes
1answer
19 views

Can every power series be representated as a taylor series?

Can every power series be represented as a Taylor series? More concrete: Given an arbitrary power series $\sum_{n=0}^\infty a_n (x-x_0)^n$, is there always a $C^\infty$-function $f$ such that ...
0
votes
1answer
147 views

I have a question about integrating, and what to do about the constant. $\displaystyle\int\frac{1}{1-z}dz$

http://www.maa.org/sites/default/files/pdf/upload_library/2/Kalman-2013.pdf On page 44, they conclude that $g'(z) = - \displaystyle\frac{\ln(1-z)}{z}$ by saying that it is just ...
3
votes
2answers
312 views

Logarithm Series: Symbol Manipulation Proof that $\log(x) + \log(y) = \log (xy)$

Let $R$ be a ring with 1. Define a formal power series $$\log(x)=\sum_{m=1}^\infty (-1)^{m+1}\frac{(x-1)^m}{m}.$$ I would like to show using only manipulations of the power series (pretending we know ...
0
votes
1answer
37 views

Calculus II: Radius of Convergence

I have this math problem that states: In each part, write out the first four terms of the series, and then find the radius of convergence. $$(a) ...
0
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1answer
30 views

Taylor polynomials of degree n

I have this math question that states: Find the Taylor polynomials of degree $n$ approximating $ln(1+x)$ for $x$ near $0$. The $n$'s are 5, 7, and 9. $f^{(5)}(0)=24$; I got the derivative to ...